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Solve the system by substitution. { 2 x 3 y = 12 −12 y + 8 x = 48

infinitely many solutions

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Solve the system by substitution. { 5 x + 2 y = 12 −4 y 10 x = −24

infinitely many solutions

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Look back at the equations in [link] . Is there any way to recognize that they are the same line?

Let’s see what happens in the next example.

Solve the system by substitution. { 5 x 2 y = −10 y = 5 2 x

Solution

The second equation is already solved for y , so we can substitute for y in the first equation.

Substitute x for y in the first equation. .
Replace the y with 5 2 x . .
Solve for x . .
.

Since 0 = −10 is a false statement the equations are inconsistent. The graphs of the two equation would be parallel lines. The system has no solutions.

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Solve the system by substitution. { 3 x + 2 y = 9 y = 3 2 x + 1

no solution

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Solve the system by substitution. { 5 x 3 y = 2 y = 5 3 x 4

no solution

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Solve applications of systems of equations by substitution

We’ll copy here the problem solving strategy we used in the Solving Systems of Equations by Graphing section for solving systems of equations. Now that we know how to solve systems by substitution, that’s what we’ll do in Step 5.

How to use a problem solving strategy for systems of linear equations.

  1. Read the problem. Make sure all the words and ideas are understood.
  2. Identify what we are looking for.
  3. Name what we are looking for. Choose variables to represent those quantities.
  4. Translate into a system of equations.
  5. Solve the system of equations using good algebra techniques.
  6. Check the answer in the problem and make sure it makes sense.
  7. Answer the question with a complete sentence.

Some people find setting up word problems with two variables easier than setting them up with just one variable. Choosing the variable names is easier when all you need to do is write down two letters. Think about this in the next example—how would you have done it with just one variable?

The sum of two numbers is zero. One number is nine less than the other. Find the numbers.

Solution

Step 1. Read the problem.
Step 2. Identify what we are looking for. We are looking for two numbers.
Step 3. Name what we are looking for.  Let n = the first number
Let m = the second number
Step 4. Translate into a system of equations. The sum of two numbers is zero.
.
One number is nine less than the other.
.
The system is: .
Step 5. Solve the system of
equations. We will use substitution
since the second equation is solved
for n .
Substitute m − 9 for n in the first equation. .
Solve for m . .
.
.
Substitute m = 9 2 into the second equation
and then solve for n .
.
.
.
.
Step 6. Check the answer in the problem. Do these numbers make sense in
the problem? We will leave this to you!
Step 7. Answer the question. The numbers are 9 2 and 9 2 .
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The sum of two numbers is 10. One number is 4 less than the other. Find the numbers.

The numbers are 3 and 7.

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The sum of two number is −6. One number is 10 less than the other. Find the numbers.

The numbers are 2 and −8.

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In the [link] , we’ll use the formula for the perimeter of a rectangle, P = 2 L + 2 W .

The perimeter of a rectangle is 88. The length is five more than twice the width. Find the length and the width.

Solution

Step 1. Read the problem. .
Step 2. Identify what you are looking for. We are looking for the length and width.
Step 3. Name what we are looking for. Let L = the length
   W = the width
Step 4. Translate into a system of equations. The perimeter of a rectangle is 88.
    2 L + 2 W = P
.
The length is five more than twice the width.
.
The system is: .
Step 5. Solve the system of equations.
We will use substitution since the second
equation is solved for L .

Substitute 2 W + 5 for L in the first equation.
.
Solve for W . .
.
.
.
Substitute W = 13 into the second
equation and then solve for L .
.
.
.
Step 6. Check the answer in the problem. Does a rectangle with length 31 and width
13 have perimeter 88? Yes.
Step 7. Answer the equation. The length is 31 and the width is 13.
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Questions & Answers

the diagram of the digestive system
Assiatu Reply
allimentary cannel
Ogenrwot
How does twins formed
William Reply
They formed in two ways first when one sperm and one egg are splited by mitosis or two sperm and two eggs join together
Oluwatobi
what is genetics
Josephine Reply
Genetics is the study of heredity
Misack
how does twins formed?
Misack
What is manual
Hassan Reply
discuss biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles
Joseph Reply
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Yousuf Reply
the study of living organisms and their interactions with one another and their environments
AI-Robot
the study of living organisms and their interactions with one another and their environment.
Wine
discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form
Joseph Reply
what is the blood cells
Shaker Reply
list any five characteristics of the blood cells
Shaker
lack electricity and its more savely than electronic microscope because its naturally by using of light
Abdullahi Reply
advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear
Abdullahi
cell theory state that every organisms composed of one or more cell,cell is the basic unit of life
Abdullahi
is like gone fail us
DENG
cells is the basic structure and functions of all living things
Ramadan
What is classification
ISCONT Reply
is organisms that are similar into groups called tara
Yamosa
in what situation (s) would be the use of a scanning electron microscope be ideal and why?
Kenna Reply
A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,
Hilary
cell is the building block of life.
Condoleezza Reply
what is cell divisoin?
Aron Reply
Diversity of living thing
ISCONT
what is cell division
Aron Reply
Cell division is the process by which a single cell divides into two or more daughter cells. It is a fundamental process in all living organisms and is essential for growth, development, and reproduction. Cell division can occur through either mitosis or meiosis.
AI-Robot
What is life?
Allison Reply
life is defined as any system capable of performing functions such as eating, metabolizing,excreting,breathing,moving,Growing,reproducing,and responding to external stimuli.
Mohamed
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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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